It could still be a high limit switch problem...checking with an ohm meter while a device is still wired to the system can produce deceptive results because of electrical feed backs that can make an open switch appear closed. (not seen with DC circuits but common with AC electrical controls circuits).
That leaves us somewhat inconclusive on the high limit control issue. You may want to try the use of a jumper as outlined earlier if the pressure switch tube cleaning fails (described below)
Regarding the cover over the fire box... if you mean the panel that encloses the gas valve and burner head area..... then the pressure switch is opening when you leave the panel in place, and closing when you remove the panel... that is reverse of the normal operation confirmed by the fact that when you replace the panel after the burner fires, the gas valve keeps running. We need to verify that by using a jumper. I will detail that later.
This could also be a defect in the control card...an intermittent defect. That is difficult to confirm even on site with instruments... we are into dangerous territory here.
It doesn't make a lot of sense... often however the small rubber tubes that run to the pressure switch get condensate in them, or insects or dirt, and the tiny orifices in the tube attachment nipples can clog preventing normal operating pressures from being active and that could cause some problems with the pressure switch. Do that cleaning first.
testing the pressure switch function with a jumper. If there are just two wire terminals on the pressure switch. you can test the function this way.
remove the wire from one of the terminals, try to start the furnace. it should not start, there should be no power to the glow plug.
Next cycle the power to the furnace to reset it, then fit a jumper across the two terminals on the pressure switch within 2 seconds of the combustion air fan starting...the igniter should energize 15 to 30 seconds later. (that delay is called the purge cycle) if the furnace works that way, the chances that the pressure switch is the problem are high.. this is all still a bit of an anomally, it could still be a problem with the control card.
REMOVE THE JUMPER AFTER THIS BRIEF TESTING.
Use a paper clip wire to poke into each of those tiny ports where the hoses attach to the furnace and to the pressure switch.... it could also be a leaking diaphragm in the pressure switch or burnt switch contacts.
If the switch sticks closed at the end of a heating cycle, and stays stuck closed next time the combustion air fan comes on, the programming on the control card will not let the furnace fire, the pressure switch needs to be *open *before the combustion air fan starts.. then close within a few seconds *after the combustion air fan starts in order for the control card program to recognize the pressure switch closing as a sign of a slight vacuum in the fire box,
All of this function needs to be verified on site by a licensed HVAC service person. These are critical safety devices.
Let me know how that goes, we can proceed from there.